The distributions of the product and ratio of random variables find
an important place in the literature and much work is done when the
random variables are independent and come from a particular probability
distribution.

If the random variables [X.sub.1], [X.sub.2], ..., [X.sub.n] are
arranged in ascending order of magnitudes and then written as
[X.sub.(1)] [less than or equal to] [X.sub.(2)],...[less than or equal
to] [X.sub.(n)], then [X.sub.(i)] is called the [i.sup.th] order
statistics (i=1,2, ...,n) and the ordered random variables are
necessarily dependent. The distribution of product and quotient of the
extreme order statistics and that of consecutive order statistics are
useful in ranking and selection problems. Subrahmaniam (16) has made the
study of product and quotient of order statistics from uniform
distribution and exponential distribution, whereas Malik and Trudel (14)
studied the cases when the order statistics are from Pareto, power and
Weibull distributions.Recently the author (21) has studied order
statistics from Kumaraswami distribution.

The subject of order statistics has been further generalized and
the concept of generalized order statistics is introduced and studied by
Kamps in a series of papers and books (26), (27), (28), (29). The order
statistics, record values and sequential order statistics are special
cases of generalized order statistics. This concept is widely studied by
many research workers namely Ahsanullah (17), (18), (19), (20),
El-Baset, Ahmed and Al-Matrofi (2), Cramer and Kamps (7), (8), Cramer,
Kamps and Rychlik (9), (10), (11), (12), Kamps and Cramer (29) and
Reiess (25).

In the present paper we shall obtain the joint distribution,
distribution of product and distribution of ratio of two generalized
order statistics from the family of distributions known as Kumaraswamy
distribution (24).

2. Definitions

(i) Generalized Order Statistics

Let F(x) denote an absolutely continuous distribution function with
density function f(x) and [X.sub.1;n,m,k], [X.sub.2;n,m,k],...,
[X.sub.n;n,m,k] (k[less than or equal to]1, m is a real number) be
'n' generalized order statistics. Then the joint probability
density function (p.d.f.) [f.sub.1, ..., n] ([x.sub.1], ..., [x.sub.n])
can be written as (26)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where,

[[gamma].sub.j] = k + (n-j)(m + 1) and f(x) = [[dF(x)]/[dx]]

If m=0 and k=1 it gives the joint p.d.f. of 'n' ordinary
order statistics [X.sub.1,n] [less than or equal to] ... [less than or
equal to] [X.sub.n,n]. If m = -1 and k = 1, it gives the joint p.d.f. of
the first 'n' upper records of the independent and identically
distributed random variables. Various distributional properties of
generalized order statistics are studied by Kamps (27) and that of
record values by Ahsanullah (17), (19), Arnold, Balakrishnan and
Nagaraja (4) and Raqab (24).

Further integrating out [x.sub.1], ..., [x.sub.[r-1]],
[x.sub.[r+1]], ..., [x.sub.n] from (1), we get p.d.f. [f.sub.r,n,m,k] of
[X.sub.r;n,m,k] (26) as

The result (6), on taking m=0 and k=1 reduces to the joint p.d.f.
of ith and jth order statistics as given in David (13).

(ii) Kumaraswamy Distribution

In this distribution, the probability density function of a random
variable X is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

with the cumulative density function (or distribution function),
given as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

In probability theory Kumaraswamy's double bounded
distribution is as versatile as the beta distribution, but much simpler
to use especially in simulation studies as it has a simple closed form
for both the p.d.f. and c.d.f.

(iii) The Mellin Transform

Let ([X.sub.1], [X.sub.2]) be a two dimensional random variable
having the joint probability density function f([x.sub.1], [x.sub.2])
that is positive in the first quadrant and zero elsewhere. The Mellin
transform of f([x.sub.1], [x.sub.2]) is defined by Fox (5) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

with the inverse

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

under the appropriate conditions discussed by Fox.

In this paper, we are interested in the following two particular
cases [16].

If Y = [X.sub.1] [X.sub.2], then h(y), the p.d.f. of Y, has the
Mellin transform

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

and if Z = [X.sub.1]/[X.sub.2], then g(z), the p. d. f. of Z, has
the Mellin transform

The definition of the H-function given by (13) will however have
meaning even if some of these quantities are zero, giving us in tern
simple transformation formulas. The nature of contour L, a set of
sufficient conditions for the convergence of this integral, the
asymptotic expansion, some of its properties and special cases can be
referred to in the book by Srivastava, Gupta and Goyal (15).

3. Joint Distribution and Distributions of Product and Ratio of Two
Generalized Order Statistics

Theorem 1. Let [X.sub.i;n,m,k] and [X.sub.j;n,m,k] be ith and jth
generalized order statistics with (i<j), based on a random sample of
size n from the Kumaraswamy distribution. The joint p.d.f of these
generalized order statistics is given by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

provided that a, b > 0, 1 [less than or equal to] i < j [less
than or equal to] n, m is a real number, k [greater than or equal to] 1
and n > 1, 0 [less than or equal to] [x.sub.i] < [x.sub.j] [less
than or equal to] 1 and [C.sub.j] and [[gamma].sub.j] are defined by
(3).

Proof. The result can easily be established on substituting the
values of f(x), F(x), and [g.sub.m](x) from equations (7), (8) and (4)
respectively in the equation (6) and expressing the values of
[[[g.sub.m](F([X.sub.i]))].sup.[i-1]] and
[[[g.sub.m](F([X.sub.j]))-[g.sub.m](F([X.sub.i]))].sup.[j-i-1]] in their
series forms.

Theorem 2. Let [X.sub.i;n,m,k] and [X.sub.j;n,m,k] denote the ith
and jth generalized order statistics from a random sample of size
'n' drawn from Kumaraswamy distribution defined by (7), then
the probability density function of the product

Y = [X.sub.i;n,m,k][X.sub.j;n,m,k] (16)

and the ratio

Z = [[X.sub.i;n,m,k]/[X.sub.j;n,m,k]] (17)

are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)

where H[z] is the Fox H-function defined by (14) and j-i +
[l.sub.1]-[l.sub.2] >0, a > 0, b>0, 1[less than or equal to]i
< j[less than or equal to]n, k[greater than or equal to]1, m and k
are real numbers and the symbols [[gamma].sub.j] and [C.sub.j] are
defined by (3).

Proof. To find the p.d.f. of the product Y, we take double Mellin
transform of eq.(15) and evaluate the integrals with the help of known
result [1, p.311, eq.(31)]. Now using (11) we obtain Mellin transform of
g(y) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

where [beta][a,b] is usual beta function. Taking inverse Mellin
transform of above equation w.r.t. 's' and using a known
result [22, p.115] we arrive at (23).

To obtain the p.d.f. of the ratio i.e. [h.sub.i,j;n,m,k](z), we use
(12) with (9) and (15) to get Mellin transform of h(z) as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Taking Mellin inversion of the above result and interpretating with
the help of definition of H-function given by (13), we get the desired
result (19).

4. Special Cases

Corollary 1. If we take a = b = 1 in Theorems 1 and 2, we get the
joint p.d.f. and p.d.f. of product and ratio of ith and jth generalized
order statistics from uniform distribution. The results are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)

where [G.sub.[2,2].sup.[1,1]][z] is Meijer G function [3].

Corollary 2. If we take j = i + 1 in Theorems 1 and 2, we get the
joint distribution and distribution of product and ratio of consecutive
generalized order statistics based on a random sample of size n from the
Kumaraswamy distribution and are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (26)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

Corollary 3. If we take i = 1, j = n in Theorems 1 and 2, we get
the joint distribution and distribution of product and ratio of the
extreme generalized order statistics based on a random sample of size n,
from Kumaraswamy distribution, and are given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (28)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (29)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Corollary 4. If we take n to be odd say 2p+1 then putting i = p+1
and j = 2p+1 in Theorem 2. we get the p.d.f. of the product and ratio of
peak to median of a random sample of size 2p+1 of generalized order
statistics as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)

Remark. If we take m = 0 and k = 1 in Theorems 1 and 2, then
generalized order statistics reduces into order statistics and we get
the joint distribution and distribution of product and ratio of order
statistics [X.sub.i,n] and [X.sub.n,n] from a sample of size n from
Kumaraswamy distribution as obtained recently by the author (21).

Acknowledgements

The author is thankful to anonymous referee for useful suggestions
which led to the present form of the paper.